A Program of The Actuarial Foundation. Aligned with Common Core State and NCTM Standards.

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What is an actuary? An actuary is an expert in statistics who works with businesses, governments, and organizations to help them plan for the future. Actuarial science is the discipline that applies math and statistical methods to assess risk.

Lesson 1: Probability Basics

In this lesson, students use the flipping of a coin to understand the relationship between probability and real-word outcomes. They also practice using diagrams, tables, and the fundamental counting principle to calculate probability.

STANDARDS (CCSS AND NCTM)

Grade 7: Statistics and Probability (CCSS 7.SP.5 and 7)

Grades 6–8: Make Sense of Problems, Construct Viable Arguments, Model with Mathematics, and Look for and Make Use of Structure (CCSS MP1, 3, 4, and 7); NCTM Data Analysis and Probability

OBJECTIVE

Students will understand that probability can be expressed as a fraction, a decimal, or a percentage; and

Students will use tree diagrams, tables, and the fundamental counting principle to calculate probability.

1. Show the class a coin. Ask the class whether it will land on heads or tails if flipped. Students should answer that it could land on either heads or tails. Ask if there is a way to quantify the chance that it will land on heads. If the class doesn't mention the word probability, introduce it and note that it means "a fraction, decimal, or percentage describing the likelihood of an event occurring." Explain how no event can have less than a 0% chance or more than a 100% chance of occurring.

2. Ask for examples of how probability is used in the real world. The topic of weather forecasts may be mentioned. Make sure the class understands that a 40% probability of precipitation means that there is a 40% likelihood that precipitation will fall within a given area. Gaming/odds may also be mentioned. You can also introduce to students the fact that companies (insurance and financial companies in particular), statistical experts such as actuaries, and individuals in daily life use probability to make reasonable predictions about the future and to assess risk.

3. Ask what the probability is of a flipped coin landing on heads (1/2). Ensure that the class understands that the numerator (1) represents the number of favorable outcomes (heads) while the denominator (2) represents all possible outcomes (heads and tails). If students haven't mentioned it, ensure that they are also able to express the probability as .5 or 50%.

4. Ask what outcomes (i.e., heads/tails combinations) are possible if the coin is flipped two times, e.g., heads/heads or tails/heads. If students begin to offer outcomes in a haphazard order, ask them how they could make sure they recorded all possible outcomes without double counting. Suggest, for example, a tree diagram and model how it can be used to identify the four different outcomes. Ask what the probability is for any one outcome (1/4, .25, or 25%).

5. Model for students how a table or an organized list could be used to determine the number of possible outcomes. See poster for example if necessary.

6. Ask what the probability is of flipping one head and one tail. There are four possible outcomes and two favorable ones (heads/tails and tails/heads). Point out that even though this would initially be depicted as 2/4, we would want to reduce to lowest terms, so our final answer would be 1/2, .5, or 50%.

7. Guided Practice: In pairs or individually, ask students to determine how many outcomes are possible if the coin were flipped three times (eight outcomes). Ask students to be prepared to explain how they used a tree diagram, a table, an organized list, or some other method to solve the problem in an organized way. Ask students to share their solution methods.

8. Ask students if there is a way to determine the number of outcomes without using a table or a tree diagram. Demonstrate how the fundamental counting principle could be used, i.e., two possible outcomes for the first, second, and third flips or 2 x 2 x 2 = 8 or 23.

9. Guided Practice: In pairs or individually, ask students to determine the number of outcomes for four flips, using a solution method of their choice. Ask students to share how they arrived at their answer. (24 = 16)

10. Independent Practice: Distribute Worksheet 1. Ensure that students are comfortable using all three probability methods.

LESSON EXTENSION
Have students use the Online Probability Challenge to practice using probability skills for real-life purposes. This interactive online activity challenges students to use probability to help Rick and Athena plan a summer concert tour. This activity can be used as an in-class lesson activity or an out-of-the-classroom extension.